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Popular Trigonometry >

30=15cos((2pi)/(365)t)+43

  • Pre Algebra
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Solution

30=15cos(3652π​t)+43

Solution

t=2π365⋅2.61927…​+365n,t=−2π365⋅2.61927…​+365n
+1
Degrees
t=8718.00664…∘+20912.95952…∘n,t=−8718.00664…∘+20912.95952…∘n
Solution steps
30=15cos(3652π​t)+43
Switch sides15cos(3652π​t)+43=30
Move 43to the right side
15cos(3652π​t)+43=30
Subtract 43 from both sides15cos(3652π​t)+43−43=30−43
Simplify15cos(3652π​t)=−13
15cos(3652π​t)=−13
Divide both sides by 15
15cos(3652π​t)=−13
Divide both sides by 151515cos(3652π​t)​=15−13​
Simplifycos(3652π​t)=−1513​
cos(3652π​t)=−1513​
Apply trig inverse properties
cos(3652π​t)=−1513​
General solutions for cos(3652π​t)=−1513​cos(x)=−a⇒x=arccos(−a)+2πn,x=−arccos(−a)+2πn3652π​t=arccos(−1513​)+2πn,3652π​t=−arccos(−1513​)+2πn
3652π​t=arccos(−1513​)+2πn,3652π​t=−arccos(−1513​)+2πn
Solve 3652π​t=arccos(−1513​)+2πn:t=2π365arccos(−1513​)​+365n
3652π​t=arccos(−1513​)+2πn
Multiply both sides by 365
3652π​t=arccos(−1513​)+2πn
Multiply both sides by 365365⋅3652π​t=365arccos(−1513​)+365⋅2πn
Simplify
365⋅3652π​t=365arccos(−1513​)+365⋅2πn
Simplify 365⋅3652π​t:2πt
365⋅3652π​t
Multiply fractions: a⋅cb​=ca⋅b​=3652⋅365π​t
Cancel the common factor: 365=t⋅2π
Simplify 365arccos(−1513​)+365⋅2πn:365arccos(−1513​)+730πn
365arccos(−1513​)+365⋅2πn
Multiply the numbers: 365⋅2=730=365arccos(−1513​)+730πn
2πt=365arccos(−1513​)+730πn
2πt=365arccos(−1513​)+730πn
2πt=365arccos(−1513​)+730πn
Divide both sides by 2π
2πt=365arccos(−1513​)+730πn
Divide both sides by 2π2π2πt​=2π365arccos(−1513​)​+2π730πn​
Simplify
2π2πt​=2π365arccos(−1513​)​+2π730πn​
Simplify 2π2πt​:t
2π2πt​
Divide the numbers: 22​=1=ππt​
Cancel the common factor: π=t
Simplify 2π365arccos(−1513​)​+2π730πn​:2π365arccos(−1513​)​+365n
2π365arccos(−1513​)​+2π730πn​
Cancel 2π730πn​:365n
2π730πn​
Cancel 2π730πn​:365n
2π730πn​
Divide the numbers: 2730​=365=π365πn​
Cancel the common factor: π=365n
=365n
=2π365arccos(−1513​)​+365n
t=2π365arccos(−1513​)​+365n
t=2π365arccos(−1513​)​+365n
t=2π365arccos(−1513​)​+365n
Solve 3652π​t=−arccos(−1513​)+2πn:t=−2π365arccos(−1513​)​+365n
3652π​t=−arccos(−1513​)+2πn
Multiply both sides by 365
3652π​t=−arccos(−1513​)+2πn
Multiply both sides by 365365⋅3652π​t=−365arccos(−1513​)+365⋅2πn
Simplify
365⋅3652π​t=−365arccos(−1513​)+365⋅2πn
Simplify 365⋅3652π​t:2πt
365⋅3652π​t
Multiply fractions: a⋅cb​=ca⋅b​=3652⋅365π​t
Cancel the common factor: 365=t⋅2π
Simplify −365arccos(−1513​)+365⋅2πn:−365arccos(−1513​)+730πn
−365arccos(−1513​)+365⋅2πn
Multiply the numbers: 365⋅2=730=−365arccos(−1513​)+730πn
2πt=−365arccos(−1513​)+730πn
2πt=−365arccos(−1513​)+730πn
2πt=−365arccos(−1513​)+730πn
Divide both sides by 2π
2πt=−365arccos(−1513​)+730πn
Divide both sides by 2π2π2πt​=−2π365arccos(−1513​)​+2π730πn​
Simplify
2π2πt​=−2π365arccos(−1513​)​+2π730πn​
Simplify 2π2πt​:t
2π2πt​
Divide the numbers: 22​=1=ππt​
Cancel the common factor: π=t
Simplify −2π365arccos(−1513​)​+2π730πn​:−2π365arccos(−1513​)​+365n
−2π365arccos(−1513​)​+2π730πn​
Cancel 2π730πn​:365n
2π730πn​
Cancel 2π730πn​:365n
2π730πn​
Divide the numbers: 2730​=365=π365πn​
Cancel the common factor: π=365n
=365n
=−2π365arccos(−1513​)​+365n
t=−2π365arccos(−1513​)​+365n
t=−2π365arccos(−1513​)​+365n
t=−2π365arccos(−1513​)​+365n
t=2π365arccos(−1513​)​+365n,t=−2π365arccos(−1513​)​+365n
Show solutions in decimal formt=2π365⋅2.61927…​+365n,t=−2π365⋅2.61927…​+365n

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